No. 3.1 MINOR PLANETS— LEUSCHNER, GLANCY, LEVY. 93 



Midtiplying the second by cos 4', the third by sin (/- and adding: 

 0,^(«,-M>u.) =-</., + |[(l-e cos b)(^W,-\e^W, + \e^J]^ + 2{[Y,] cos ^ + [Z,] sin ^0 



— (1 — « cos b){w+ l^i)^ I { ^'j cos 4' + Z.i sin (,'' — [I'j cos ^'' + ^2 sin ipDde 



-[(1 -f cos e)j{T,-[T,])chJ^ 



in which 



■"2 = [2/3] cos ^ + [2,] sin (;{' 



and [A'J, [Y^], [Z^] are read by inspection from T,, which is to be determined as follows: 

 If Z 50, eqs. (89), (90), are written in the form 



[1 — cos (/-co)] = 4 1 — 27J cos £ — cos (s — ^) +)} cos (2s — (/-) +ri cos (/-+ 



cos^ <p a^ cos^ (p 



^ T + 3"^^ 2 [l-co3(/-c.i)]| = 3 + 14 f-8Ticoss + 2f cos 2 s-2 cos (s-^'') 



cos-" (p[ a- a' cos' 9> 



— 8 ]j^ cos (s — ^) + 2?) cos (2s — 4') + 2rj cos ^ + 



then TgF and T',, given by Z 49, eqs. (84), (85), in connection with Z 50, eq. (87), are given by 



T-^= — Tj — 4{1— 2?) cos £ — cos (s — 0) + tj cos (2s — tl') + ij cos 9!- + } 



ISp. q{n + r. -n + s)rjPTj'^j-' sin ^ 



T', = {3 + 14ij' - 8)j cos £ + 2i}' cos 2s - 2 cos (s - ^) - Stj^ cos (s - ^'0 + 2)} cos (2s - ^) + 2jj cos ^ 



+ }ISj,. ,{n + r.-n + s)7jPri'9i-( sinA-^H-iv)^ 



and r, (Table XVIIIa) is computed by Z 53, eq. (94), in which 



E = x + 2riii, Et = x, + 2riyt 

 The function 



<t>2-'u<' = [X:,]-Tj[y2] 



is tabulated in Table XXI; the function 



Wj = [.V2] cos 4' + [^2] sin (p 

 is tabulated in Table XXII. 



From the latter [j/j] can be read by inspection, and 7j[;/j] added to the former gives [zj]. 



Finally, (Table XXIIa), 



f TT,] = [Xj] + [:/,] cos 4' + [22] sin 4 



